I've always found questions like this (a) very interesting, but (b) kind of hard to think about. I'm going to try to verbalize some inchoate thoughts on the subject. One part of a question like this is kind of easy to answer. How "unlikely" is the specific observation being reported? Obviously the chance of the first 8 digits being even is a little less than 2^-8, a bit under a half a percent. So that seems reasonably surprising. The other part is the hard part, the imponderable part. I start by observing that whether Jim "should be" surprised, he certainly _was_ surprised -- at least to the extent necessary to make him ask the question he did. How many different possible "surprising" observations of this kind are there? This is a psychology question, not a math question, and I have no idea how to go about answering it. But it's crucial to this situation. Suppose there were a thousand potentially surprising observations, each with a probability of around half a percent. Then it would be surprising (not sure in what sense I mean this) if none of them came true, wouldn't it? For example: we expect the nth Fibonacci number to have around n log[10] phi digits. If the nth Fibonacci number starts with (n log[10] phi) / 2 even digits, that would be surprising (in the naive sense). I am confident of this because F91 didn't even clear that bar, but still surprised Jim. Anyway, there is clearly some infinite series whose sum tells us the chance of this surprise occuring for _any_ n >= 90 -- I think it's comparable to a geometric series, so it must converge. I have no idea what that number is. But all the events being measured are certainly capable of "surprising" Jim (and me, I admit). If the series sums to 1/3, then what should we conclude? And this is considering only potential observations that are clearly so closely related to Jim's observation of F91 that they would obviously elicit similar emotions. But what other observations could have done that? I don't know how to start answering that questien. On Fri, Apr 26, 2019 at 2:35 PM James Propp <jamespropp@gmail.com> wrote:

The 91st Fibonacci number is

4,660,046,610,375,530,309

Should I be surprised that the first eight digits are even?

Should I be surprised that, if you strip out the zero-digits, this number consists of six even digits followed by eight odd digits?

I know that the initial digits of the Fibonacci numbers have an asymptotic equidistribution property, but 91 is not that big...

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